For \({{A}}, {{B}}, {{C}}\in \mathbb {R}^{ N \times N}\) , we let \(|{{A}}|\)  denote the matrix having entries \(|{{A}}_{i,j}|\) .
For the matrices \({{B}}, {{C}}\)  we define \({{B}}\le {{C}}\)  to mean that \({{B}}_{i,j}\le {{C}}_{i,j}\)  for each \(i\)  and \(j\) .
The following is a folklore result in linear algebra (e.g. see [1]}, [2]}).
